Do you or do you not agree with the formulas written by ANGELO MARGARIS in his book ,FIRST ORDER MATHEMATICAL LOGIC ,pages 45 to 46??If YOU DO ,there is nothing to be said ,because i do agree with him, except that PLATO'S formulas are wrong.IF YOU DO NOT, then any arguments should be against ANGELO MARGARIS.

Having taught courses from Margaris’ book, even though it was years ago, I would say that I well understand the points he is making.
I think that it is you who do not understand the logic here.

On page 45, the instructions for #1 say “Find a useful denial…”The statement in part (e) is the symbolic statement for continuity; whereas the statement in part (f) is not.
The negation given on page 46 is exactly the one I gave above without any implication.
This is a denial of a universal giving an existential which does not contain an implication.

Having taught courses from Margaris’ book, even though it was years ago, I would say that I well understand the points he is making.
I think that it is you who do not understand the logic here.

On page 45, the instructions for #1 say “Find a useful denial…”The statement in part (e) is the symbolic statement for continuity; whereas the statement in part (f) is not.
The negation given on page 46 is exactly the one I gave above without any implication.
This is a denial of a universal giving an existential which does not contain an implication.

The formula in part (e) of the question in Margari's book is:

The formula in part (f) of the question in the same book is:

Which is the formula that i wrote in my post # 21

Now if the formula (e) is the right one ,what is the formula (f) for.

Besides ,when we start a proof for continuity at a point ,we start with the assumption:

Let ε>0, a feature which is part of formula (f) and not of formula (e).

IN A FORMAL PROOF for continuity at a point ,only formula (f) is applicable and not formula (e).

HENCE the right formula is (f) and its negation is the formula that i wrote in my post #21

However if you still insist that the right formula is (e) ,you have to prove that .

In a formal proof ,where everything is written down one can easily see that the right formula is (f),whilst in an ordinary mathematical proof the whole issue can be confused.

For example in formaly proving that the simple function,f(x) =2x+3 is continuous at any point c we can only use formula (f)

This thread is getting out of hand, so I'm going to close it. Remember everyone, SHOUTING and typing in bold doesn't make what you say correct, it's just impolite. Let's all be courteous to each other.